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Quadric surfaces in standard form practice problems

Rewrite a quadric surface in standard form and name it. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

7 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 9.641

easy
Write \( \displaystyle - 9 x^{2} + 36 y^{2} - 36 z = 0 \) in standard form and identify the surface.
  1. Divide through by 36 and solve for the variable to the first power.
    Reviewed
  2. \[ \frac{- 9 x^{2} + 36 y^{2} - 36 z}{36} = - \frac{x^{2}}{4} + y^{2} - z \]
    The standard form.✓ Proved
  3. A hyperbolic paraboloid: z to the first power equals a difference of squares.
    Reviewed
Answer \( - \frac{x^{2}}{4} + y^{2} - z = 0\ (\text{hyperbolic paraboloid}) \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.642

easy
Write \( \displaystyle 225 x^{2} + 900 y^{2} + 36 z^{2} = 900 \) in standard form and identify the surface.
  1. Divide through by 900.
    Reviewed
  2. \[ \frac{225 x^{2} + 900 y^{2} + 36 z^{2}}{900} = \frac{x^{2}}{4} + y^{2} + \frac{z^{2}}{25} \]
    The standard form.✓ Proved
  3. A ellipsoid: x²/a² + y²/b² + z²/c² = 1.
    Reviewed
Answer \( \frac{x^{2}}{4} + y^{2} + \frac{z^{2}}{25} = 1\ (\text{ellipsoid}) \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.643

easy
Write \( \displaystyle 72 x^{2} + 72 y^{2} - 2 z^{2} = 72 \) in standard form and identify the surface.
  1. Divide through by 72.
    Reviewed
  2. \[ \frac{72 x^{2} + 72 y^{2} - 2 z^{2}}{72} = x^{2} + y^{2} - \frac{z^{2}}{36} \]
    The standard form.✓ Proved
  3. A hyperboloid of one sheet: two squares positive and one negative, = 1.
    Reviewed
Answer \( x^{2} + y^{2} - \frac{z^{2}}{36} = 1\ (\text{hyperboloid of one sheet}) \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.644

easy
Write \( \displaystyle 4 x^{2} + 4 y^{2} - 144 z = 0 \) in standard form and identify the surface.
  1. Divide through by 144 and solve for the variable to the first power.
    Reviewed
  2. \[ \frac{4 x^{2} + 4 y^{2} - 144 z}{144} = \frac{x^{2}}{36} + \frac{y^{2}}{36} - z \]
    The standard form.✓ Proved
  3. A elliptic paraboloid: z (or another variable) to the first power equals a sum of squares.
    Reviewed
Answer \( \frac{x^{2}}{36} + \frac{y^{2}}{36} - z = 0\ (\text{elliptic paraboloid}) \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.645

easy
Write \( \displaystyle 9 x^{2} + 36 y^{2} - z^{2} = 36 \) in standard form and identify the surface.
  1. Divide through by 36.
    Reviewed
  2. \[ \frac{9 x^{2} + 36 y^{2} - z^{2}}{36} = \frac{x^{2}}{4} + y^{2} - \frac{z^{2}}{36} \]
    The standard form.✓ Proved
  3. A hyperboloid of one sheet: two squares positive and one negative, = 1.
    Reviewed
Answer \( \frac{x^{2}}{4} + y^{2} - \frac{z^{2}}{36} = 1\ (\text{hyperboloid of one sheet}) \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.651

easy
Write \( \displaystyle - 16 x^{2} - 4 y^{2} + 16 z^{2} = 144 \) in standard form and identify the surface.
  1. Divide through by 144.
    Reviewed
  2. \[ \frac{- 16 x^{2} - 4 y^{2} + 16 z^{2}}{144} = - \frac{x^{2}}{9} - \frac{y^{2}}{36} + \frac{z^{2}}{9} \]
    The standard form.✓ Proved
  3. A hyperboloid of two sheets: one square positive and two negative, = 1.
    Reviewed
Answer \( - \frac{x^{2}}{9} - \frac{y^{2}}{36} + \frac{z^{2}}{9} = 1\ (\text{hyperboloid of two sheets}) \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.794

easy
Write \( \displaystyle 9 x^{2} + 36 y^{2} + z^{2} = 36 \) in standard form and identify the surface.
  1. Divide through by 36.
    Reviewed
  2. \[ \frac{9 x^{2} + 36 y^{2} + z^{2}}{36} = \frac{x^{2}}{4} + y^{2} + \frac{z^{2}}{36} \]
    The standard form.✓ Proved
  3. A ellipsoid: x²/a² + y²/b² + z²/c² = 1.
    Reviewed
Answer \( \frac{x^{2}}{4} + y^{2} + \frac{z^{2}}{36} = 1\ (\text{ellipsoid}) \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More quadric surfaces in standard form practice

Write \( \displaystyle 288 x^{2} + 800 y^{2} - 450 z^{2} = 7200 \) in standard form and identify the surface.
Problem 9.646easy✓ Nihil obstat
Write \( \displaystyle 3 x^{2} + 27 y^{2} - 27 z = 0 \) in standard form and identify the surface.
Problem 9.647easy✓ Nihil obstat
Write \( \displaystyle 4 x^{2} + 9 y^{2} - 36 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.648easy✓ Every equation proved
Write \( \displaystyle 25 x^{2} + 36 y^{2} - 100 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.649easy✓ Every equation proved
Write \( \displaystyle 81 x^{2} + 144 y^{2} - 81 z^{2} = 1296 \) in standard form and identify the surface.
Problem 9.650easy✓ Nihil obstat
Write \( \displaystyle x^{2} + 4 y^{2} - z^{2} = 0 \) in standard form and identify the surface.
Problem 9.652easy✓ Every equation proved
Write \( \displaystyle 18 x^{2} + 288 y^{2} - 32 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.653easy✓ Every equation proved
Write \( \displaystyle 225 x^{2} + 324 y^{2} - 8100 z = 0 \) in standard form and identify the surface.
Problem 9.654easy✓ Nihil obstat
Write \( \displaystyle 72 x^{2} + 18 y^{2} - 8 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.655easy✓ Every equation proved
Write \( \displaystyle 2 x^{2} + 2 y^{2} - 2 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.726easy✓ Every equation proved
Write \( \displaystyle 18 x^{2} + 32 y^{2} + 72 z^{2} = 288 \) in standard form and identify the surface.
Problem 9.727easy✓ Every equation proved
Write \( \displaystyle 2 x^{2} + 2 y^{2} - 8 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.728easy✓ Every equation proved
Write \( \displaystyle 75 x^{2} + 12 y^{2} - 75 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.729easy✓ Every equation proved
Write \( \displaystyle 72 x^{2} + 50 y^{2} + 200 z^{2} = 1800 \) in standard form and identify the surface.
Problem 9.730easy✓ Nihil obstat
Write \( \displaystyle - 27 x^{2} + 75 y^{2} - 675 z = 0 \) in standard form and identify the surface.
Problem 9.731easy✓ Every equation proved
Write \( \displaystyle - 3 x^{2} + 75 y^{2} - 75 z = 0 \) in standard form and identify the surface.
Problem 9.732easy✓ Every equation proved
Write \( \displaystyle 400 x^{2} + 100 y^{2} - 16 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.733easy✓ Every equation proved
Write \( \displaystyle - 225 x^{2} + 9 y^{2} - 225 z = 0 \) in standard form and identify the surface.
Problem 9.734easy✓ Every equation proved
Write \( \displaystyle 8 x^{2} + 2 y^{2} - 72 z = 0 \) in standard form and identify the surface.
Problem 9.735easy✓ Nihil obstat
Write \( \displaystyle - 50 x^{2} - 32 y^{2} + 50 z^{2} = 800 \) in standard form and identify the surface.
Problem 9.736easy✓ Nihil obstat
Write \( \displaystyle - 3 x^{2} + 3 y^{2} - 12 z = 0 \) in standard form and identify the surface.
Problem 9.737easy✓ Nihil obstat
Write \( \displaystyle - 18 x^{2} + 50 y^{2} - 450 z = 0 \) in standard form and identify the surface.
Problem 9.738easy✓ Every equation proved
Write \( \displaystyle - 4 x^{2} + y^{2} - 16 z = 0 \) in standard form and identify the surface.
Problem 9.739easy✓ Every equation proved
Write \( \displaystyle - 16 x^{2} + 36 y^{2} - 576 z = 0 \) in standard form and identify the surface.
Problem 9.740easy✓ Nihil obstat
Write \( \displaystyle - 144 x^{2} + 4 y^{2} - 144 z = 0 \) in standard form and identify the surface.
Problem 9.787easy✓ Nihil obstat
Write \( \displaystyle 64 x^{2} + 100 y^{2} - 1600 z^{2} = 1600 \) in standard form and identify the surface.
Problem 9.788easy✓ Nihil obstat
Write \( \displaystyle - 200 x^{2} - 450 y^{2} + 72 z^{2} = 1800 \) in standard form and identify the surface.
Problem 9.789easy✓ Nihil obstat
Write \( \displaystyle 72 x^{2} + 50 y^{2} - 50 z^{2} = 1800 \) in standard form and identify the surface.
Problem 9.790easy✓ Nihil obstat
Write \( \displaystyle 16 x^{2} + 100 y^{2} - 400 z = 0 \) in standard form and identify the surface.
Problem 9.791easy✓ Nihil obstat
Write \( \displaystyle 450 x^{2} + 50 y^{2} - 18 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.792easy✓ Every equation proved
Write \( \displaystyle - 75 x^{2} + 12 y^{2} - 300 z = 0 \) in standard form and identify the surface.
Problem 9.793easy✓ Every equation proved
Write \( \displaystyle 900 x^{2} + 2025 y^{2} + 324 z^{2} = 8100 \) in standard form and identify the surface.
Problem 9.795easy✓ Every equation proved
Write \( \displaystyle 36 x^{2} + 36 y^{2} - 9 z^{2} = 144 \) in standard form and identify the surface.
Problem 9.796easy✓ Nihil obstat
Write \( \displaystyle 2 x^{2} + 72 y^{2} - 8 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.829easy✓ Every equation proved
Write \( \displaystyle - 16 x^{2} - y^{2} + 4 z^{2} = 16 \) in standard form and identify the surface.
Problem 9.830easy✓ Every equation proved
Write \( \displaystyle 75 x^{2} + 108 y^{2} - 108 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.831easy✓ Every equation proved
Write \( \displaystyle 12 x^{2} + 12 y^{2} - 3 z^{2} = 12 \) in standard form and identify the surface.
Problem 9.832easy✓ Nihil obstat
Write \( \displaystyle 50 x^{2} + 18 y^{2} - 18 z^{2} = 450 \) in standard form and identify the surface.
Problem 9.833easy✓ Nihil obstat
Write \( \displaystyle - 3 x^{2} + 108 y^{2} - 108 z = 0 \) in standard form and identify the surface.
Problem 9.834easy✓ Nihil obstat
Write \( \displaystyle 8 x^{2} + 8 y^{2} + 2 z^{2} = 72 \) in standard form and identify the surface.
Problem 9.835easy✓ Nihil obstat
Write \( \displaystyle 1296 x^{2} + 2025 y^{2} - 900 z^{2} = 32400 \) in standard form and identify the surface.
Problem 9.836easy✓ Nihil obstat
Write \( \displaystyle 1296 x^{2} + 2025 y^{2} + 3600 z^{2} = 32400 \) in standard form and identify the surface.
Problem 9.837easy✓ Nihil obstat
Write \( \displaystyle 675 x^{2} + 432 y^{2} - 1200 z^{2} = 10800 \) in standard form and identify the surface.
Problem 9.838easy✓ Nihil obstat
Write \( \displaystyle - 36 x^{2} - 25 y^{2} + 25 z^{2} = 900 \) in standard form and identify the surface.
Problem 9.900easy✓ Nihil obstat
Write \( \displaystyle - 675 x^{2} - 300 y^{2} + 432 z^{2} = 10800 \) in standard form and identify the surface.
Problem 9.901easy✓ Nihil obstat
Write \( \displaystyle 18 x^{2} + 8 y^{2} - 288 z = 0 \) in standard form and identify the surface.
Problem 9.902easy✓ Nihil obstat
Write \( \displaystyle 3600 x^{2} + 2025 y^{2} - 1296 z^{2} = 32400 \) in standard form and identify the surface.
Problem 9.903easy✓ Nihil obstat
Write \( \displaystyle 81 x^{2} + 9 y^{2} - 9 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.904easy✓ Every equation proved
Write \( \displaystyle - 75 x^{2} + 108 y^{2} - 2700 z = 0 \) in standard form and identify the surface.
Problem 9.905easy✓ Nihil obstat
Write \( \displaystyle 108 x^{2} + 27 y^{2} - 12 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.906easy✓ Every equation proved
Write \( \displaystyle 8 x^{2} + 18 y^{2} - 18 z^{2} = 0 \) in standard form and identify the surface.
Problem 9.907easy✓ Every equation proved
Write \( \displaystyle 4 x^{2} + 16 y^{2} - 16 z = 0 \) in standard form and identify the surface.
Problem 9.908easy✓ Every equation proved
Write \( \displaystyle 2 x^{2} + 72 y^{2} + 18 z^{2} = 72 \) in standard form and identify the surface.
Problem 9.909easy✓ Every equation proved